Monitor capture doesn't record sound, it captures the image that shows on your monitor, nothing more. Sound is recorded via the two sound channels that OBS has defined, typically one for desktop sound, and one for a microphone.
I don't really know what the sound from your monitor means, does your monitor play sound? Also, you don't want the video from your monitor to be streamed, then how does streaming the video game work?
If you want what I think you want, to stream a game, but only include certain programs' sound in the stream, you could take a look at this tutorial perhaps:
http://obsproject.com/forum/viewtopic.php?f=18&t=3202